3.1 Bad conditioning THE ASTEROID IDENTIFICATION PROBLEM 2.3 Restricted orbit identification

   
3. Problems with the linear theory

The linear theory for orbit identification provides the most rigorous mathematical setting to solve this problem. A mathematical theorem, however, is not at all a rigorous tool unless all the hypothesis are applicable to the concrete problem to be solved. The hypothesis needed to apply the formalism of Section 2 are the following:

1.

The normal matrices $C_1, C_2$ and $C_0=C_1+C_2$ are invertible and positive-definite.

2.

The map between the space of the residuals $\xi$ and the space of estimated parameters $X$ is in the linear regime, that is the linearized map $\Delta X= -\Gamma\,B^T\xi$ is a good approximation in a region including both orbits.

3.

The observation errors are of a size consistent with the residual normalization adopted.

In this Section we shall discuss the applicability of these three hypothesis to the problem of orbit identification.

 

• 3.1 Bad conditioning
• 3.2 Nonlinearity
• 3.3 Normalization/weighting of the residuals